Kinetic theory gives the most fundamental model of low temperature gas discharges
[1]}, [2]}. We assume a plane-parallel geometry with Cartesian coordinates.
The \(x\) -axis points from the electrode at the position \(x=0\)  to the bulk. The opposite electrode is located at \(x=L\) . Translational invariance in \(y\)  and \(z\)  as well as axisymmetry around the \(x\) -axis are assumed.
To avoid having to define unphysical boundary conditions at the sheath-plasma transition, we treat the whole discharge
domain \([0,L]\)  and select as the sheath edge \(x_\mathrm {B}\)  the first point that is always quasineutral in the RF period.
The ion density at the point \(x_\mathrm {B}\)  will be called \(n_\mathrm {B}\) . (The slight arbitrariness in these definitions is harmless.)
The discharge is periodic in the RF period \(T_\mathrm {RF} = 2\pi \omega _\mathrm {RF}\) .
Kinetic equations are formulated for the distribution functions \(f_s(x,v_\parallel ,v_\perp ,t)\) ,
\(s= 1\ldots N_\mathrm {s}\) .
In our numerical examples below, \(N_\mathrm {s}=2\)  with \(\mathrm {e} \equiv 1\)  and \(\mathrm {i}\equiv 2\) .We introduce cylindrical coordinates in velocity space, with \(v_\parallel \equiv v_x\)  and \(v_\perp = \sqrt{v_y^2+v_z^2}\) .
Moreover, we abbreviate \(E \equiv E_x\) .
The term on the right is the collision term to represent elastic and inelastic collisional interaction and chemistry:
\(\frac{\partial f_s}{\partial t} + v_\parallel \frac{\partial f_s}{\partial x} +\frac{q_s}{m_s} E \frac{\partial f_s}{\partial v_\parallel } = \frac{\partial f_s}{\partial t}\Bigl \vert _\mathrm {c}.\) 
